Problem: $P(x)$ is a polynomial. $P(x)$ divided by $(x+7)$ has a remainder of $5$. $P(x)$ divided by $(x+3)$ has a remainder of $-4$. $P(x)$ divided by $(x-3)$ has a remainder of $6$. $P(x)$ divided by $(x-7)$ has a remainder of $9$. Find the following values of $P(x)$. $P(-3)=$
Solution: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, $P({-3})$ is equal to the remainder when $P(x)$ is divided by $(x-({-3}))$, which can be rewritten as $(x+3)$, and we are given that this remainder is equal to $-4$. In a similar manner, $P({7})$ is equal to the remainder when $P(x)$ is divided by $(x-{7})$, and we are given that this remainder is equal to $9$. In conclusion, $P(-3)=-4$ $P(7)=9$